Graphing parabolas | Year 9

01 / LEARN

The idea, step by step

A quadratic graph is a parabola

The graph of $y = x^{2}$ is a smooth U-shaped curve called a parabola. Substitute several $x$ values, plot the resulting ordered pairs, then draw a smooth curve through them. Do not join the points with sharp straight-line corners.

For a rule such as $y = x^{2} - 4$ , squaring a negative value still gives a positive result. For $x = - 3$ , calculate ${(- 3)}^{2} - 4 = 9 - 4 = 5$ .

Values for $y = x^{2} - 4$
x	-3	-2	-1	0	1	2	3
y	5	0	-3	-4	-3	0	5

Read the main features

An x-intercept is where the graph crosses or touches the x-axis, so $y = 0$ there. The corresponding $x$ values are the roots of the quadratic equation.

The turning point is the lowest point of an upward-opening parabola, or the highest point of a downward-opening one. The axis of symmetry is the vertical line through the turning point.

y = x^{2} - 4 : turning point (0, - 4), axis x = 0

Compare supplied rules

In $y = {(x - h)}^{2} + k$ , the turning point is $(h, k)$ . For example, $y = {(x - 2)}^{2} - 3$ has turning point $(2, - 3)$ . This reads a supplied rule; you do not need to complete the square.

A negative sign in front of the square turns the parabola upside down. Multiplying the square by $2$ doubles each height measured from the turning point, making the shape narrower. Adding a constant moves the graph up or down.

Graphing software can help test several parameter values. Change one parameter at a time, predict the effect, then compare the graph with your prediction.

y = - {(x - 2)}^{2} + 3 : turning point (2, 3), opens down

The parabola y = x² - 4

Explore a parabola

Change one value at a time. Describe what moves, what changes shape and what stays the same.

y = x²

a = 1h = 0k = 0

Turning point: (0, 0). Opens upwards.

A common trap

Saying $y = {(x - 2)}^{2} + 1$ has turning point $(- 2, 1)$ .

Instead: The square is zero when $x = 2$ . The turning point is $(2, 1)$ , and the axis of symmetry is $x = 2$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Make a table

For $y = x^{2} - 1$ , find the points for $x = - 2, - 1, 0, 1, 2$ .

Substitute each value using brackets for negatives. The $y$ values are $3, 0, - 1, 0, 3$ .
Plot $(- 2, 3), (- 1, 0), (0, - 1), (1, 0), (2, 3)$ and draw a smooth U-shaped curve.

The turning point is

(0, - 1)

and the roots are

- 1

and

1

Example 2

Read a shifted parabola

For $y = {(x - 3)}^{2} - 4$ , state the turning point and axis of symmetry, then find the roots.

Read the turning point as $(3, - 4)$ , so the axis is $x = 3$ .
At an x-intercept, $0 = {(x - 3)}^{2} - 4$ , so ${(x - 3)}^{2} = 4$ .
Then $x - 3 = 2$ or $x - 3 = - 2$ , giving $x = 5$ or $x = 1$ .

Turning point

(3, - 4)

; axis

x = 3

; roots

1

and

5

Example 3

Read a downward-opening graph

For $y = - x^{2} + 9$ , state its highest point and its x-intercepts.

The negative sign means the graph opens down. At $x = 0$ , $y = 9$ , giving its highest point $(0, 9)$ .
Set $y = 0$ : $x^{2} = 9$ , so $x = - 3$ or $x = 3$ .

Highest point

(0, 9)

; x-intercepts

(- 3, 0)

and

(3, 0)

Check your understanding

For $y = {(x + 1)}^{2} - 4$ , state the turning point and roots.

Show answer

Turning point $(- 1, - 4)$ ; roots $- 3$ and $1$ .

The square is zero at $x = - 1$ . At an x-intercept, ${(x + 1)}^{2} = 4$ , so $x + 1 = 2$ or $- 2$ .

03 / WORKSHEET

Your turn

6 questions · 21 marks

Worksheet PDF Worked solutions

Read the questions online

For $y = x^{2} - 4$ , calculate $y$ for $x = - 3, - 2, - 1, 0, 1, 2, 3$ . Write the ordered pairs.
3 marks
Use the graph of $y = x^{2} - 4$ supplied with this worksheet.
1. (a)State the turning point.
2. (b)State the x-intercepts.
3. (c)Write the axis of symmetry.
4 marks
The parabola y = x² - 4
For $y = {(x - 2)}^{2} - 9$ :
1. (a)State the turning point and whether the graph opens up or down.
2. (b)Find the roots.
4 marks
For $y = - x^{2} + 9$ :
1. (a)State the highest point.
2. (b)Find both roots.
3. (c)State whether its turning point is a maximum or minimum.
4 marks
Compare $y = x^{2}$ and $y = 2 x^{2}$ .
1. (a)Find the $y$ value on each graph when $x = 1$ .
2. (b)Which graph is narrower?
3 marks
Compare $y = x^{2} + 3$ and $y = x^{2} - 2$ .
1. (a)How far above the second graph is the first graph?
2. (b)Do they have the same axis of symmetry? Give its equation.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

For $y = x^{2} - 4$ , calculate $y$ for $x = - 3, - 2, - 1, 0, 1, 2, 3$ . Write the ordered pairs.
3 marks
Correct values for negative inputs: $5, 0, - 3$ [1].
Correct value for zero: $- 4$ [1].
Correct values for positive inputs: $- 3, 0, 5$ and ordered pairs in $(x, y)$ order [1].
$(- 3, 5), (- 2, 0), (- 1, - 3), (0, - 4), (1, - 3), (2, 0), (3, 5)$ .
Use the graph of $y = x^{2} - 4$ supplied with this worksheet.
1. (a)State the turning point.
2. (b)State the x-intercepts.
3. (c)Write the axis of symmetry.
4 marks
(a) Turning point $(0, - 4)$ [1].
(b) Give the x-intercept $(- 2, 0)$ [1] and $(2, 0)$ [1].
(c) Axis $x = 0$ [1].
1. (a) $(0, - 4)$ .
2. (b) $(- 2, 0)$ and $(2, 0)$ .
3. (c) $x = 0$ .
For $y = {(x - 2)}^{2} - 9$ :
1. (a)State the turning point and whether the graph opens up or down.
2. (b)Find the roots.
4 marks
(a) Turning point $(2, - 9)$ [1]; opens up [1].
(b) Use ${(x - 2)}^{2} = 9$ , so $x - 2 = \pm 3$ [1]; both roots $- 1, 5$ [1].
1. (a) $(2, - 9)$ ; opens up.
2. (b) $x = - 1$ or $x = 5$ .
For $y = - x^{2} + 9$ :
1. (a)State the highest point.
2. (b)Find both roots.
3. (c)State whether its turning point is a maximum or minimum.
4 marks
(a) Highest point $(0, 9)$ [1].
(b) Give the root $- 3$ [1] and $3$ [1].
(c) Maximum [1].
1. (a) $(0, 9)$ .
2. (b) $- 3$ and $3$ .
3. (c)A maximum.
Compare $y = x^{2}$ and $y = 2 x^{2}$ .
1. (a)Find the $y$ value on each graph when $x = 1$ .
2. (b)Which graph is narrower?
3 marks
(a) $y = 1$ for $y = x^{2}$ [1]; $y = 2$ for $y = 2 x^{2}$ [1].
(b) $y = 2 x^{2}$ is narrower [1].
1. (a) $1$ and $2$ .
2. (b) $y = 2 x^{2}$ .
Compare $y = x^{2} + 3$ and $y = x^{2} - 2$ .
1. (a)How far above the second graph is the first graph?
2. (b)Do they have the same axis of symmetry? Give its equation.
3 marks
(a) Find $3 - (- 2) = 5$ units [1].
(b) State they share an axis [1] and give $x = 0$ [1].
1. (a) $5$ units.
2. (b)Yes; $x = 0$ .

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Curriculum reference & lesson scope

AC9M9A06: Connect quadratic rules and graphs and explore parameter changes

A hand-worked introduction connecting AC9M9A04 graphs and roots with AC9M9A06 parameter changes. Students read supplied turning-point forms and compare simple rules. AC9M9A06 also requires digital experimentation; this reading and worksheet alone do not assess that full competency.

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